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The vector sum of two forces is perpendicular to their vector differences. In that case, the forces :
A
Are equal to each other
B
Are equal to each other in magnitude
C
Are not equal to each other in magnitude
D
Cannot be predicted
Detailed Solution
Two vectors are perpendicular if their dot product is zero.
$(\vec{A} + \vec{B})\cdot(\vec{A} - \vec{B}) = 0$
$\Rightarrow A^2 - \vec{A}\cdot\vec{B} + \vec{B}\cdot\vec{A} - B^2 = 0$
Since $\vec{A}\cdot\vec{B} = \vec{B}\cdot\vec{A}$, the middle terms cancel: $A^2 - B^2 = 0$
$\Rightarrow A = B$
So the forces are equal in magnitude. They need not be equal as vectors, since their directions can be different (if they were equal vectors, their difference would be zero).
$(\vec{A} + \vec{B})\cdot(\vec{A} - \vec{B}) = 0$
$\Rightarrow A^2 - \vec{A}\cdot\vec{B} + \vec{B}\cdot\vec{A} - B^2 = 0$
Since $\vec{A}\cdot\vec{B} = \vec{B}\cdot\vec{A}$, the middle terms cancel: $A^2 - B^2 = 0$
$\Rightarrow A = B$
So the forces are equal in magnitude. They need not be equal as vectors, since their directions can be different (if they were equal vectors, their difference would be zero).